Instability of periodic travelling waves with mean zero for a 1D Boussinesq system

We consider herein the instability properties of the periodic traveling wave solutions of a general nonlinear Boussinesq system related with a dispersive model for the 1D propagation of nonlinear long water waves with small amplitude, via an adaptation of the result of M. Grillakis, J. Shatah, and W. Strauss for systems with a special Hamiltonian structure. In a particular case of this general system, we use Jacobian elliptic functions to build a curve of L-periodic traveling wave solutions having mean zero in [0,L] and also verify the validity of the criteria used to establish instability, in a specific range of the wave speed. Furthermore, we provide numerical evidence on a type of instability arising when perturbing with small amplitude disturbances by using a highlyaccurate spectral numerical scheme.

Keywords

dispersive equations, periodic traveling-waves, cnoidal, snoidal, waves, orbital instability

2010 Mathematics Subject Classification

35Q51, 35Q53, 76B25

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Published 23 July 2012